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Floquet Hamiltonian and Entanglement in Spin Systems in Periodic Magnetic Fields

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Abstract

We investigate entanglement in a two-spin system with the dipole–dipole interaction in the multi-pulse spin locking nuclear magnetic resonance (NMR) experiment. We discover a conflict between an exact solution and the standard approximation employing a time-independent effective Floquet Hamiltonian. While the exact solution shows nonzero entanglement in the system, the perturbative Floquet approach predicts that entanglement is absent. The failure of the Floquet method is explained by the multivaluedness of the Floquet Hamiltonian. Correct results can only be obtained with a proper choice of the branch of the Hamiltonian. The same issue is present in other applications of the Floquet theorem in the perturbation theory beyond NMR.

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Notes

  1. Note that the unitary transformation (52) is local and hence cannot change entanglement [32]. Entanglement changes when an approximate effective time-independent Hamiltonian is produced from Eq. (52). Indeed, the averaging of the Hamiltonian over the period \(2\pi /\omega _1\) can be understood as a nonlocal transformation [N. N. Bogoliubov and Yu. A. Mitropolskiy, Asymptotic Methods in the Theory of Nonlinear Oscillations (Gordon and Breach, New York, 1964) and Ref. [20]] which affects entanglement. This explains the failure of the perturbative Floquet approach.

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Authors and Affiliations

  1. Institute of Problems of Chemical Physics of RAS, Chernogolovka, Moscow Region, 142432, Russia

    E. B. Fel’dman & E. I. Kuznetsova

  2. Department of Physics, Brown University, Providence, RI, 02912, USA

    D. E. Feldman

Authors
  1. E. B. Fel’dman
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  2. D. E. Feldman
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  3. E. I. Kuznetsova
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Correspondence to E. I. Kuznetsova.

Additional information

This work was supported by the Russian Foundation for Basic Research (Grant 16-03-00056) and the Program of the Presidium of RAS “Electron spin resonance, spin-dependent electron effects, and spin technologies” (Grant 0089-2015-0191).

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Fel’dman, E.B., Feldman, D.E. & Kuznetsova, E.I. Floquet Hamiltonian and Entanglement in Spin Systems in Periodic Magnetic Fields. Appl Magn Reson 48, 517–531 (2017). https://doi.org/10.1007/s00723-017-0879-8

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